Is "$(\exists z)$ the sky is blue" a proposition? I am unsure as this sentence certainly states something which could be either true or false; however the $(\exists z)$ seems meaningless here- I am unsure if the fact that $z$ is unused after it has been quantified is relevant here.
Thanks
It is a proposition. Here the predicate $P(z)$ would be the statement "the sky is blue." Although this has nothing to do with $z$, it is still a valid predicate in $z$ (analogous to a constant map from one set to another).
Does there exist some object $z$ such that the sky is blue? Why yes, no matter what object $z$ you pick, the sky is still blue. So in fact, we have the more general proposition: "$(\forall z)\text{ the sky is blue.}$"
Of course, this is assuming the sky really is always blue...
You’re referring to vacuous quantification. While it is kind of odd, vacuous quantification is innocuous/redundant at the level of syntax, and it simplifies things at the level of semantics.
That is, $P$, $\exists x P$, and $\forall x P$ are all equivalent for a formula $P$ with $x$ not free. Vacuous quantification then allows us to more easily define what constitutes a well-formed formula (wff); namely, if $P$ is a formula and $x$ a variable, then $\forall x P$ and $\exists x P$ are formulas.
